非线性二阶微分方程的交换分解：理论与应用
Commutative Factorization of Nonlinear Second-Order Differential Equations: Theory and Applications
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中文总结 AI 辅助
本文推广交换分解框架至二阶非线性常微分方程，结合Riccati-Bernoulli方程与Bäcklund变换，系统构造特解与通解，并在经典力学及非线性振荡模型中验证其有效性。
中文摘要 AI 辅助
本工作针对一大类二阶非线性常微分方程，提出了交换分解框架的推广。主要目标是扩展交换分解过程，以便利用Riccati-Bernoulli方程和Bäcklund变换。这为构造原始二阶非线性方程的特定解和通解提供了一条系统途径。理论发展通过几个具有物理和数学意义的非线性模型加以说明，包括经典力学和非线性振荡系统中出现的方程。结果表明，交换分解构成了解非线性微分方程的有效分析工具。
英文摘要
This work presents a generalization of the commutative factorization framework for a broad class of second-order nonlinear ordinary differential equations. The main objective was to extend the commutative factorization procedure in order to use the Riccati-Bernoulli equation and the Bäcklund transformation. This provides a systematic route for constructing both particular and general solutions of the original second-order nonlinear equation. The theoretical development is illustrated through several nonlinear models of physical and mathematical interest, including equations arising in classical mechanics and nonlinear oscillatory systems. The results demonstrate that commutative factorization constitutes an effective analytical tool for solving nonlinear differential equations.
发表机构
- Universidad de Guadalajara（瓜达拉哈拉大学）
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